Growth of face-homogeneous tessellations

Abstract

A tessellation of the plane is face-homogeneous if for some integer k ≥ 3 there exists a cyclic sequence σ = [p 0 , p 1 , . . . , p k−1 ] of integers ≥ 3 such that, for every face f of the tessellation, the valences of the vertices incident with f are given by the terms of σ in either clockwise or counter-clockwise order. When a given cyclic sequence σ is realizable in this way, it may determine a unique tessellation (up to isomorphism), in which case σ is called monomorphic, or it may be the valence sequence of two or more non-isomorphic tessellations (polymorphic). A tessellation whose faces are uniformly bounded in the hyperbolic plane but not uniformly bounded in the Euclidean plane is called a hyperbolic tessellation. Such tessellations are well-known to have exponential growth. We seek the face-homogeneous hyperbolic tessellation(s) of slowest growth rate and show that the least growth rate of such monomorphic tessellations is the "golden mean," γ = (1 + √ 5)/2, attained by the sequences [4, 6, 14] and [3, 4, 7, 4]. A polymorphic sequence may yield non-isomorphic tessellations with different growth rates. However, all such tessellations found thus far grow at rates greater than γ.

The Golden Section's heptagonal connections

“…Most likely they were not new even back in 2003 when I first obtained them as a spin-off from a paper concerned with certain types of fullerene graphs. Since then, several papers appeared addressing some related but more general problems [4,6,13], and most of the present results can be obtained by specializing certain parameters in the main results of references [7][8][9]. However, the results are not widely known either -only a handful of the enumerating sequences are present in the On-Line Encyclopedia of Integer Sequences, and mostly without references to the corresponding tessellations.…”

Infinite Graphical Frobenius Representations

“…The family of GFRs to be presented in this section consists of bipartite chiral planar maps of valence at least 6. Since all faces have even covalence at least 4, these maps accrue faces too fast to admit uniformly bounded faces in the Euclidean metric (see [4]); they have vertex-homogeneous embeddings in the hyperbolic plane and their growth is exponential.…”

“…Infinite, almost transitive graphs of connectivity 1 other than the double ray (which was discussed in Subsection 5.1) have precisely this tree-like structure. When such graphs are vertex-transitive, every vertex is a cut-vertex incident either with at least three lobes 4 or with at least two lobes of which at least one is biconnected. As their automorphism groups must likewise have exponential growth, we might rightly expect our GFRs to be graphical representations involving free products of groups.…”

Tiling Rectangles and the Plane Using Squares of Integral Sides

“…Problems of finding or computing special arrangements such as tiling, packing, stacking, tessellation, and Box-Rectangular Drawings of Planar Graphs [1,2] are categorized in the computational geometry field, with many applications in industry, science, and technology. Although most of the applications of tiling in image processing are related to different shapes of pixels and uniform polygons, tiling with non-uniform shapes also has many applications in this field.…”